Depth of modular invariant rings

It is well-known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be Cohen-Macaulay and computing the depth is often very difficult. In this paper¹ we obtain a simple formula for the depth of the ring of invariants for a family of modular representations. This family includes all modular representations of cyclic groups. In particular, we obtain an elementary proof of the celebrated theorem of Ellingsrud and Skjelbred [6].     

Differents in modular invariant theory

Let be a graded polynomial algebra over a field k, such that each variable is homogeneous of positive degree. No restrictions are made with respect to the field. Let the finite group G act on A by graded algebra automorphisms and denote the subalgebra of invariants by B. In this paper the various "different ideals" of the extension are studied that define the ramification locus. We prove, for example, that the subring of invariants is itself a polynomial ring if and only if the ramification locus is pure of height one. Here the ramification locus is defined by either the Kahler different, the Noether different or the Galois different. As a consequence we prove that the invariant ring is itself a polynomial ring if and only if there are invariants whose Jacobian determinant does not vanish and is of degree δ, where δ is the degree of the Dedekind different. Using this criterion we give a quick proof of Serre's result that if the invariant ring is a polynomial algebra, then the group is generated by generalized reflections.     

A relative cohomological invariant for group pairs

We define a cohomological invariantE(G, S, M) whereG is a group,S is a non empty family of (not necessarily distinct) subgroups of infinite index inG andM is a -module ( is the field of two elements). In this paper we are interested in the special case where the family of subgroups consists of just one subgroup, andM is the -module . The invariant will be denoted byE(G, S). We study the relations of this invariant with other endse(G), e(G, S) ande(G,S)), and some results are obtained in the case whereG andS have certain properties of duality.     

On Saito-Kurokawa lifting to cohomological Siegel modular forms

We discuss a theta lifting from to O(3,2), which will produce a certain class of residual cohomological automorphic forms on the orthogonal group. We will show an explicit formula for their Fourier expansions, in which the constant terms may also occur, by using the Fourier coefficients of a half-integral weight cusp form, which is similar to the classical formula for the holomorphic Saito-Kurokawa lifting at finite places.Mathematics Subject Classification (2000): 11F46, 11F27, 11F30     

Circular method and modular theory

A close relation between the classical circular method and modern modular theory is shown by the example of binary problems in number theory. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 3–5. Translated by B. M. Bekker     

Ghosts in modular representation theory

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finite-dimensional G-representations factor through a projective—we define the ghost number of kG to be the smallest integer l such that the composite of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C₂ as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.     

Public-key cryptography and invariant theory

We suggest public-key cryptosystems based on groups invariants. We also give an overview of known cryptosystems that involve groups. Bibliography: 33 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 293, 2002, pp. 26–38.This revised version was published online in April 2005 with a corrected cover date and article title.     

Frobenius splitting and invariant theory

We consider varieties over an algebraically closed field k of characteristicp>0. Given a linear representation of a reductive group, we prove that the ring of invariants is F-regular provided the associated projective quotient is Frobenius-split, the twisting sheaves are Cohen-Macaulay (C-M), and a mild technical condition is met. As an example of how this can be used, we show that the ring of invariants (under the adjoint action of SL (3)) ofg copies ofM ₃ is C-M. (HereM ₃ denotes the vector space of 3×3 matrices over k andp>3.) The method of proof involves an induction, and is potentially of wide applicability. As a corollary we obtain that the moduli space of rank 3 and degree 0 bundles on a smooth projective curve of genusg is C-M.     

Trace polynomials and invariant theory

LetA be a finite-dimensional simple (non-associative) algebra over an algebraically closed fieldF of characteristic 0. LetG be the group of its automorphisms which acts onkA, the direct sum ofk copies ofA. SupposeA is generated byk elements. In this paper, generators of the field of rational invariantF(kA) ^G are described in terms of operations of the algebraA.     

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

We consider purely inseparable extensions of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group and a vector space decomposition such that and , where denotes the integral closure. Moreover, is Cohen-Macaulay if and only if is Cohen-Macaulay. Furthermore, is polynomial if and only if is polynomial, and if and only if

where and .

     

Noncommutative classical invariant theory

In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspace? _d ^m of the noncommutative invariant algebra? _d consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S _m . We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(? _d,t) whose coefficients are the number of irreducibleS _m -modules in the decomposition of? _d ^m , and show that it is rational. Finally, we develop some analogues of all this for covariants.     

Noncommutative classical invariant theory

In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspaceĨ _d ^m of the noncommutative invariant algebraĨ _d consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S _m . We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(Ĩ _d,t) whose coefficients are the number of irreducibleS _m -modules in the decomposition ofĨ _d ^m , and show that it is rational. Finally, we develop some analogues of all this for covariants.     

Knot theory and the Casson invariant in Artin presentation theory

In Artin presentation theory, a smooth, compact four-manifold is determined by a certain type of presentation of the fundamental group of its boundary. Topological invariants of both three-and four-manifolds can be calculated solely in terms of functions of the discrete Artin presentation. González-Acuña proposed such a formula for the Rokhlin invariant of an integral homology three-sphere. This paper provides a formula for the Casson invariant of rational homology spheres. Thus, all 3D Seiberg-Witten invariants can be calculated by using methods of the theory of groups in Artin presentation theory. The Casson invariant is closely related to canonical knots determined by an Artin presentation. It is also shown that any knot in any three-manifold appears as a canonical knot in Artin presentation theory. An open problem is to determine 4D Seiberg-Witten and Donaldson invariants in Artin presentation theory.     

Descent principle in modular Galois theory

We propound a descent principle by which previously constructed equations over GF(q _n)(X) may be deformed to have incarnations over GF(q)(X) without changing their Galois groups. Currently this is achieved by starting with a vectorial (= additive)q-polynomial ofq-degreem with Galois group GL(m, q) and then, under suitable conditions, enlarging its Galois group to GL(m, q _n) by forming its generalized iterate relative to an auxiliary irreducible polynomial of degreen. Elsewhere this was proved under certain conditions by using the classification of finite simple groups, and under some other conditions by using Kantor’s classification of linear groups containing a Singer cycle. Now under different conditions we prove it by using Cameron-Kantor’s classification of two-transitive linear groups.